Finding the Shortest Ladder: Solving for the Optimal Angle

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In summary, the problem involves finding the shortest ladder that goes over a fence and reaches a wall that is one meter away. The equation for the ladder is (3sqrt(3) / sin x) + sec x, where x is the angle the ladder makes with the ground and the wall. To find the shortest ladder, the derivative of the ladder equation is taken and set equal to zero, and then solved for x.
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alpha01
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Finding the "shortest ladder"

Problem

I was given an analogy involving a ladder that goes over a fence and then leans against a wall a meter after the fence. The question wanted me to answer "what is the shortest ladder that goes over the fence and reaches the wall"

The Attempt at a Solution



I worked out the equation of the "ladder" (i.e line from the ground to the wall) is:

(3sqrt(3) / sin x) + sec x

(where x is the angle the ladder makes with the ground and the wall)

How can i work out the "shortest" ladder that can be made to reach the wall. Is is simply the derivative of the ladder?
 
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So 3sqrt(3) must be the height of the ladder h, right? Yes, the length of the ladder is L=h/sin(x)+1/cos(x). Differentiate that with respect to the angle, x, set it equal to zero and solve for x.
 

1. How can I find the shortest ladder for a specific height?

To find the shortest ladder for a specific height, you will need to measure the height of the area you need to reach. Then, you can use the Pythagorean theorem to calculate the length of the ladder needed. The formula is a² + b² = c², where a is the height of the area and b is the distance from the base of the ladder to the wall. Once you have the length, you can search for ladders with that specific length or slightly longer.

2. Can I use a ladder with a lower length than the calculated one?

No, it is not recommended to use a ladder with a lower length than the calculated one. This could result in the ladder being too short for the height needed, leading to instability and potential accidents. It is always best to use a ladder that meets or slightly exceeds the calculated length.

3. What factors should I consider when choosing a ladder for a specific height?

When choosing a ladder for a specific height, you should consider the material, weight capacity, and safety features of the ladder. The material should be sturdy and able to support the weight of the user. The weight capacity should also be able to support the weight of the user and any equipment or materials being carried on the ladder. Additionally, the ladder should have safety features such as anti-slip feet and handrails to prevent accidents.

4. Are there any other methods for finding the shortest ladder?

Yes, there are other methods for finding the shortest ladder. One method is to use a ladder with adjustable height, such as a telescoping ladder, to fit the specific height needed. Another method is to use a ladder with an adjustable angle, such as a multi-position ladder, to reach different heights with the same ladder.

5. How can I ensure the ladder I choose is safe for use?

To ensure the ladder you choose is safe for use, you should follow the manufacturer's instructions and safety guidelines. Make sure the ladder is set up on a flat and stable surface, and the weight capacity is not exceeded. Inspect the ladder for any damage or defects before each use. It is also important to use proper climbing techniques and have someone hold the ladder for extra stability if needed.

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